# Clarify on Birkhoff's theorem (in universal algebra)

In class we said that, as a consequence of Birkhoff's theorem, the theory of fields is not axiomatizable only by equations. In particular, we saw that the product of two fields in general is just a ring. However I don't understand why if we use a language with $$(+,\cdot,-,\frac {1}{}, 1,0)$$, where $$+$$ and $$\cdot$$ are the binary operations of sum and product, $$-$$ and $$\frac 1 {}$$ are the unary operations that give the inverses, and $$1,0$$ are the two usual constants. It seems to me that in this language the theory of fields is axiomatizable adding to the axioms of a ring (that are all equations) the axiom $$\frac 1 x \cdot x=1$$, that is an equation too. Probably I'm missing some hypothesis on the language in the Birkhoff's theorem, but I don't see any of them. Thaks for any clarify

The problem is that $$(x\mapsto 1/x)$$ is only a partial operation in fields (division by $$0$$ is undefined), that cannot be captured in universal algebra where all operations have to be total.